Atomic partial wave meter by attosecond coincidence metrology

Attosecond chronoscopy is central to the understanding of ultrafast electron dynamics in matter from gas to the condensed phase with attosecond temporal resolution. It has, however, not yet been possible to determine the timing of individual partial waves, and steering their contribution has been a substantial challenge. Here, we develop a polarization-skewed attosecond chronoscopy serving as a partial wave meter to reveal the role of each partial wave from the angle-resolved photoionization phase shifts in rare gas atoms. We steer the relative ratio between different partial waves and realize a magnetic-sublevel-resolved atomic phase shift measurement. Our experimental observations are well supported by time-dependent R-matrix numerical simulations and analytical soft-photon approximation analysis. The symmetry-resolved, partial-wave analysis identifies the transition rate and phase shift property in the attosecond photoelectron emission dynamics. Our findings provide critical insights into the ubiquitous attosecond optical timer and the underlying attosecond photoionization dynamics.


Supplementary Note 1: XUV-APT and attosecond photoelectron spectra
Supplementary Figures 1 (a) and (b) show the measured RABBITT trace integrated over the photoemission angles in the argon atom, and the reconstructed spectrum via FROG-CRAB through the algorithm of PCGPA 1 , respectively. The photoelectron kinetic energy in our measurements was calibrated by the ionization potential of the 3p electron of argon atom, I p = 15.76 eV. As shown in Supplementary Fig. 1(c), the pulse duration of the XUV-APT is estimated to be 320 attoseconds, the relative chirp distribution of the XUV-APT is calibrated through the effective photoionization time delay of argon, where the chirp of the H19 is set to be zero. eV, are aligned perpendicular to the polarization direction of the XUV-APT with negative asymmetry parameters, due to the relative ratio between the final s and d-partial waves [2][3][4] . However, the photoelectron PAD of the sidebands keeps following the polarization axis of the NIR pulse as Θ T varies from 0 • to 90 • 5, 6 . Supplementary Figures 2(q-x) show the PADs of argon in the same conditions. The asymmetry parameters of the main bands of argon are larger than zero with the PADs aligning along the XUV-APT polarization axis.
Supplementary (q-x) as same as (a-d) but for neon and argon results, respectively.

Supplementary Note 2: Photoionization time delay reconstruction routine
The sideband amplitude shows an oscillation as a function of the relative pump-probe time delay between the XUV-APT and the NIR pulse. As illustrated in the main text, the initial phases of the sidebands encode the dispersion of the XUV-APT and the scattering phase shifts during the photoionization process 8 . Supplementary Figure 4 shows the experimentally measured angleintegrated attosecond photoelectron spectra of the helium atom at a skew angle Θ T = 0 • . To acquire the oscillation phase and amplitude, we perform a fast Fourier transform (FFT) analysis of the RABBITT spectrum. The oscillation term around 2ω NIR can be fitted in the time domain as S(τ, θ) = a 0 cos(2ω NIR τ + ϕ 0 (θ)) + b 0 , where τ is the pump-probe time delay and b 0 is the averaged yield of a given sideband, a 0 and ϕ 0 are the oscillation amplitude and initial phase shifts. The initial phase ϕ 0 of the sideband can be decomposed into the phase difference between consecutive harmonics, ϕ XUV−APT , and the intrinsic atomic phase shifts ϕ 2hν (θ) during the photoionization process 9-13 , ϕ 0 (θ) = ϕ XUV−APT + ϕ 2hν (θ). To describe an atomic system containing N +1 electrons we divide the physical space around the nucleus into two regions. The inner region, confined to small radial distances from the nucleus, requires a full account of all electronic interactions including electron exchange which is accomplished using a configuration-interaction approach. A single electron may escape this inner region, at which point it is spatially isolated from the residual N -electron core, and electron exchange may be neglected. In the outer region, then, a single, ionized electron moves under the influence of both the long-range potential of the N -electron ion and the laser field. The full complexity of the many-body system is thus confined to the small, inner region, while the outer region is effectively a single-active electron problem. This division of space is necessary to reduce the computational load of describing the electron exchange in particular, especially for ionisation problems where the electron may travel far from the nucleus.
RMT leverages this division of space further by employing a markedly different numerical scheme in each region. In the inner region, a B-spline basis set is used to ensure accurate and efficient determination of the multielectron wavefunction. By contrast, in the outer region a grid-based finite difference scheme is employed. This also facilitates the use of a sophisticated, multi-layered parallelisation scheme, allowing RMT to be deployed on massively-parallel highperformance computers.
The N + 1-electron atom is constructed by coupling a continuum electron to the N -electron ionic states. This additional electron may occupy a bound orbital, yielding the ground and excited states, or it may be in the continuum, offering an accurate description of the ionized, N -electron ion and its interaction with the outgoing electron. This also allows the description of the outgoing wavepacket to be cast in terms of ionization 'channels', with each channel representing the emis-sion of an electron with a particular set of quantum numbers, coupled to a specific ionic state. This channel formalism is key in the present work, as the outgoing electron may be decoupled from the residual in order to provide the partial wave decomposition of the continuum wavefunction.
For both argon and neon, the atomic structure description is based on the R-matrix basis reported elsewhere 17 , and includes all outgoing electron emission channels up to maximum total angular momentum of L max = 3 (convergence of the results was checked with L max = 5). The total number of LM L Sπ symmetries included in the calculation is 31. The inner region radius is set at 20 a 0 , where a 0 is the Bohr radius. For helium this inner region radius is 15 a 0 , and the atomic structure description is the so-called '1T' description reported elsewhere 18 . This includes electron emission channels up to L max = 5 attached to the 2 S e ionisation threshold, and a single 1s orbital in the description of He + . This gives a total of 36 LM L Sπ symmetries. For all calculations, the outer region is 5160 a 0 , which is sufficiently large to ensure it is never reached by the outgoing wavepacket, which can cause non-physical reflections that obscure the resulting wavefunction. The wavefunction for the outgoing electron is obtained by decoupling it from the wavefunction for the residual ion 19 . It is subsequently transformed into momentum space through a Fourier Transform.
The laser parameters are reported in the main text, and are chosen as a compromise between the experimental values, and those which will enable numerically stable and tractable calculations. The electric field profiles of the NIR and XUV-APT pulses are shown in Supplementary Fig. 5, along with the frequency comb of the XUV-APT.

Full partial wave analysis
The partial wave transition pathways for two-photon ionization of helium are illustrated in Supplementary Table 1. Supplementary Tables 2 and 3 show the partial wave transition pathways of the neon and argon atoms. The final states in neon and argon involve several, degenerate ionic states. The final state of the coupled (ion + photoelectron) system is required to decompose the partial wave channels properly. The selection rules for the various transitions enforce that the total magnetic quantum number must not change (∆m = 0) when Θ T = 0 • and that ∆m = ±1 for Θ T = 90 • . Only for Θ T = 0 • and 90 • , can the final, total symmetry be determined exactly, and the resulting PADs can then be understood as arising from specific m-values of the outgoing electron.
For the other skew angles, the outgoing electrons with initial magnetic quantum number m i = 0 can transition to a continuum state with final magnetic quantum number m f = 0 or m f = ±1. In this case, one needs to consider the output of a full quantum calculation, and to uncouple, using the appropriate Clebsch-Gordan coefficients 20 , the outer electron from the residual ion.

Intensity dependence of the photoionization time delay in helium, neon and argon
To determine the effect of NIR intensity on the m-resolved partial wave phase shifts as a function of Θ T , we perform RMT calculations at three different NIR intensities: 1.0TW/cm 2 , 0.5TW/cm 2 and 0.1TW/cm 2 . Supplementary Figures 11(a-c) show the effective atomic phase shift as a function of the skew angle and intensity of the NIR field for the helium atom within the kinetic energy range from 3.7eV to 9.9eV. As the skew angle weighted purple lines shown in Supplementary Fig.   11(a), the effective phase shift of SB18, ϕ 2hν SB18 , varies from −0.099π to −0.107π at 1.0TW/cm 2 , whereas the ϕ 2hν SB18 maintains a constant value of −0.104π over all skew angles at the NIR intensity I NIR = 0.1TW/cm 2 , close to the perturbative limit. Supplementary Figures 13(a-c) show the effective atomic phase shifts of neon with the same NIR intensities as in helium. As Θ T increases from 0 • to 90 • , the effective atomic phase shift of SB18, ϕ 2hν SB18 , changes from −0.0720π to −0.0367π with a maximum phase shift difference of around 0.035π. This skew angle assisted phase shift difference decreases to 0.028π and 0.023π at the NIR intensity of I NIR = 0.5TW/cm 2 and 0.1TW/cm 2 , respectively.
During the XUV-NIR two-photon ionization process of neon and argon, there are three possible residual-ion states, P 0 and P ±1 , whose contributions must be summed incoherently in the final sidebands. As shown in Supplementary Fig. 14(a), each m-resolved partial wave maintains a constant phase shift over all skew angles at the perturbative limit (I NIR = 0.1 TW/cm 2 ). In the case of P 0 , ϕ 2hν We find that both p ±1 and f 0/±1/±2 partial waves present an almost identical phase shift between different residual-ion states. The major difference appears in the p 0 partial wave. We can conclude that the skew-angle-dependent p 0 partial wave phase shift variation in Supplementary  Fig. 14(c) arises from the incoherent sum over different ionic states. Since the two-photon phase shifts of p ±1 -and f 0,±1,±2 -waves only have subtle ionic-state-induced phase shift differences, the effective phase shift variation as a function of the skew angle is mainly from ϕ 2hν p 0 that changes from ϕ 2hν p 0 coupled to P 0 ionic state to ϕ 2hν p 0 coupled to P ±1 ionic states as the NIR field rotates from 0 • to 90 • .
As the NIR intensity increases from I NIR = 0.1 TW/cm 2 to I NIR = 1.0 TW/cm 2 , all mresolved partial waves show a slight variation as a function of the skew angle, and p 0 generally shows a stronger skew angle dependence than p ±1 and f 0,±1,±2 .
Supplementary Figures 14 (j-r) shows the same results but in argon, and its skew angle dependence and NIR intensity dependence is analogous to neon. However, as shown in Supplementary Figs. 14 (j-l), the partial wave phase shift in the perturbative limit is −0.119π, 0.0537π, −0.0140π and −0.0145π for p 0 , p ±1 , f 0 and f ±1 -waves coupled to the P 0 residual-ion state, and −0.0172π, −0.0188π, −0.0136π, −0.0140π and −0.0137π for p 0 , p ±1 , f 0 , f ±1 and f ±2 -waves coupled to the P ±1 residual-ion state. We find that the phase shifts of the p 0 -and p ±1 -waves are about 0.116π and 0.092π larger than those in neon in the case of P 0 residual-ion state, which can be mainly attributed to the short-range phase shift difference between δ Ne 2p→s and δ Ar 3p→s during the one-photon ionization process 23 . Supplementary Figure 15 displays the relative two-photon transition phase shifts between pand f -waves as a function of the skew angle. In comparison with Fig. 5 in main text, the ∆ϕ 2hν p 0 −f 0 and ∆ϕ 2hν p ±1 −f ±1 phases shown here are reconstructed from the experimental results by applying the partial wave proportions predicted by the SPA model.
Supplementary Figure 16, Supplementary Fig. 17, and Supplementary Fig. 18 present the photoemission angle-resolved phase shift distributions as a function of the skew angle and NIR intensity for helium, neon, and argon atoms. The observed phase shift angular distribution is from the partial wave interference, where an incoherent sum should also be considered when the initial states are degenerate.

Analytical two photon ionization model
To extract the analytical dependence on relative polarization angle, we use the 'soft photon approximation' 24 , which expresses the S-matrix element for the exchange of n NIR photons with a single XUV photon as (S1) Here, α NIR and ϵ XUV are the polarization vectors of the NIR and XUV fields respectively, k n is the sideband momentum, ϕ NIR and ϕ XUV are their respective carrier-envelope phases, Ψ i is the initial-state wave function, and χ kn is the final-state wave function.
Typically, the final state is expressed as a plane wave, but the Coulomb potential may be taken into account using the appropriate expansion where σ l = arg [Γ (l + 1 + iη)] is the Coulomb phase shift, and F l (η, kr) is the regular Coulomb function, with η = −Z/k. We align the XUV field at a relative angle Θ T to the z-axis, and assume the lowest order pathways, which for the RABBITT processes of interest involve the n = 1 (absorption) and n = −1 (emission) contributions. The angle dependence is obtained by expanding the n = ±1 Bessel functions for small arguments: so that for an arbitrary relative polarization So, the amplitudes become where ϕ 2q±1 are the carrier-envelope phases of harmonics 2q ± 1, and ψ l i m i is the initial-state wavefunction with orbital angular momentum l i and magnetic quantum number m i .
helium For helium, the angular integral is given by and so only the p 0 intermediate state contributes, so that Eq. (S14) becomes where Using Eq. (S7), the amplitudes in Eq. (S5) are given by Expressing cos θ k and sin θ k sin ϕ k as spherical harmonics, and then contracting the resulting products of spherical harmonics, Eq. (S9) may be written as The photoelectron yield for each partial wave lm is then calculated by squaring the coefficient of the spherical harmonic Y lm . From Eq. (S11), the yields of s and d 0 electrons depend only on cos 2 Θ T as and the yield of d ±1 electrons both depend on sin 2 Θ T as neon and argon For neon an argon, the XUV matrix element is given by where f 1 (r) is an initial p orbital. The angular integral is given by and so the s, d 0 , d ±1 intermediate state contribute, so that Eq. (S14) becomes and Using Eq. (S16), the amplitudes in Eq. (S5) are given by Expressing cos θ k and sin θ k sin ϕ k as spherical harmonics, and then contracting the resulting products of spherical harmonics, Eq. (S19) may be written as From Eq. (S20), the Θ T -dependent yield of f electrons is found by squaring the coefficients of the spherical harmonics Y 3m for m = 0, ±1, ±2. Note that contributions from initial states with different magnetic quantum number m i must be summed incoherently. Beginning with f 0 , the amplitude contains a term with coefficient cos Θ T originating from an initial m i = 0, and a term with coefficient sin Θ T from initial states with m i = +1 and m i = −1. These are squared and summed incoherently to give the yield of f 0 electrons, P f 0 : Continuing this procedure for all f electrons in the final state, we find that The yield of p 0 and p ±1 electrons is more complicated, and involves the relative contributions of the p → s → p and p → d → p pathways. Once again summing over different m i values incoherently, their yields are given by and P p 1 ∼ |R dp | 2 25 To determine the yields of p 0 and p 1 electrons, the coefficients of cos 2 Θ T and sin 2 Θ T are given by the photoelectron lm-resolved yields at Θ T = 0 • and Θ T = 90 • respectively. We obtain these yields from the RMT calculations, by integrating the lm-resolved momentum distributions over the photoelectron momentum and angular variables.